Aleksander Weron

Universal Algorithm for Identification of Fractional Brownian Motion. A Case of Telomere Subdiffusion

We present a systematic statistical analysis of the recently measured individual trajectories of fluorescently labeled telomeres in the nucleus of living human cells. The experiments were performed in the U2OS cancer cell line. We propose an algorithm for identification of the telomere motion. By expanding the previously published data set, we are able to explore the dynamics in six time orders, a task not possible earlier. As a result, we establish a rigorous mathematical characterization of the stochastic process and identify the basic mathematical mechanisms behind the telomere motion. We find that the increments of the motion are stationary, Gaussian, ergodic, and even more chaotic--mixing. Moreover, the obtained memory parameter estimates, as well as the ensemble average mean square displacement reveal subdiffusive behavior at all time spans. All these findings statistically prove a fractional Brownian motion for the telomere trajectories, which is confirmed by a generalized p-variation test. Taking into account the biophysical nature of telomeres as monomers in the chromatin chain, we suggest polymer dynamics as a sufficient framework for their motion with no influence of other models. In addition, these results shed light on other studies of telomere motion and the alternative telomere lengthening mechanism. We hope that identification of these mechanisms will allow the development of a proper physical and biological model for telomere subdynamics. This array of tests can be easily implemented to other data sets to enable quick and accurate analysis of their statistical characteristics.

Ergodic properties of stationary stable processes

We derive spectral necessary and sufficient conditions for stationary symmetric stable processes to be metrically transitive and mixing. We then consider some important classes of stationary stable processes: Sub-Gaussian stationary processes and stationary stable processes with a harmonic spectral representation are never metrically transitive, the latter in sharp contrast with the Gaussian case. Stable processes with a harmonic spectral representation satisfy a strong law of large numbers even though they are not generally stationary. For doubly stationary stable processes, sufficient conditions are derived for metric transitivity and mixing, and necessary and sufficient conditions for a strong law of large numbers.

Innovations and Wold decompositions of stable sequences

SummaryFor symmetric stable sequences, notions of innovation and Wold decomposition are introduced, characterized, and their ramifications in prediction theory are discussed. As the usual covariance orthogonality is inapplicable, the non-symmetric James orthogonality is used. This leads to right and left innovations and Wold decompositions, which are related to regression prediction and least pth moment prediction, respectively. Independent innovations and Wold decompositions are also characterized; and several examples illustrating the various decompositions are presented.

Fractal market hypothesis and two power-laws

Abstract A fractal approach is used to analyze financial time series by applying different degrees of time resolutions. This leads to the heterogenous market hypothesis (HMH), where different market participants analyze past events and news with different time horizons. A new general model for asset returns is studied in the framework of the fractal market hypothesis (FMH). It concerns capital market systems in which the conditionally exponential dependence (CED) property can be attached to each investor on the market.

Diffusion and relaxation controlled by tempered alpha-stable processes.

We derive general properties of anomalous diffusion and nonexponential relaxation from the theory of tempered alpha-stable processes. The tempering results in the existence of all moments of operational time. The subordination by the inverse tempered alpha-stable process provides diffusion (relaxation) that occupies an intermediate place between subdiffusion (Cole-Cole law) and normal diffusion (exponential law). Here we obtain explicitly the Fokker-Planck equation and the Cole-Davidson relaxation function. This model includes subdiffusion as a particular case.

Computer simulation of Lévy α-stable variables and processes

The aim of this paper is to demonstrate how the appropriate numerical, statistical and computer techniques can be successfully applied to the modeling of some physical systems. We propose to use a fast and accurate method of computer generation of Levy α-stable random variates.

Stable Lévy motion approximation in collective risk theory

Collective risk theory is concerned with random fluctuations of the total net assets, the risk reserve, of an insurance company. In this paper we consider weak approximations in risk theory which are especially relevant whenever the claim experience allows for heavy-tailed claims. We approximate the risk process by an α-stable Levy motion (1 < α < 2) with drift. The ruin probability within a finite time horizon is estimated. Finally, a numerical example is presented.

Harmonizable stable processes on groups: spectral, ergodic and interpolation properties

SummaryThis work extends to symmetric α-stable (SαS) processes, 1<α<2, which are Fourier transforms of independently scattered random measures on locally compact Abelian groups, some of the basic results known for processes with finite second moments and for Gaussian processes. Analytic conditions for subordination of left (right) stationarily related processes and a weak law of large numbers are obtained. The main results deal with the interpolation problem. Characterization of minimal and interpolable processes on discrete groups are derived. Also formulas for the interpolator and the corresponding interpolation error are given. This yields a solution of the interpolation problem for the considered class of stable processes in this general setting.

Guidelines for the Fitting of Anomalous Diffusion Mean Square Displacement Graphs from Single Particle Tracking Experiments

Single particle tracking is an essential tool in the study of complex systems and biophysics and it is commonly analyzed by the time-averaged mean square displacement (MSD) of the diffusive trajectories. However, past work has shown that MSDs are susceptible to significant errors and biases, preventing the comparison and assessment of experimental studies. Here, we attempt to extract practical guidelines for the estimation of anomalous time averaged MSDs through the simulation of multiple scenarios with fractional Brownian motion as a representative of a large class of fractional ergodic processes. We extract the precision and accuracy of the fitted MSD for various anomalous exponents and measurement errors with respect to measurement length and maximum time lags. Based on the calculated precision maps, we present guidelines to improve accuracy in single particle studies. Importantly, we find that in some experimental conditions, the time averaged MSD should not be used as an estimator.

FARIMA MODELING OF SOLAR FLARE ACTIVITY FROM EMPIRICAL TIME SERIES OF SOFT X-RAY SOLAR EMISSION

A time series of soft X-ray emission observed by the Geostationary Operational Environment Satellites from 1974 to 2007 is analyzed. We show that in the solar-maximum periods the energy distribution of soft X-ray solar flares for C, M, and X classes is well described by a fractional autoregressive integrated moving average model with Pareto noise. The model incorporates two effects detected in our empirical studies. One effect is a long-term dependence (long-term memory), and another corresponds to heavy-tailed distributions. The parameters of the model: self-similarity exponent H, tail index α, and memory parameter d are statistically stable enough during the periods 1977-1981, 1988-1992, 1999-2003. However, when the solar activity tends to minimum, the parameters vary. We discuss the possible causes of this evolution and suggest a statistically justified model for predicting the solar flare activity.